scholarly journals New stability theorems of uncertain differential equations with time-dependent delay

2021 ◽  
Vol 6 (1) ◽  
pp. 623-642
Author(s):  
Zhifu Jia ◽  
◽  
Xinsheng Liu
Keyword(s):  
Differential Equations ◽  
Time Dependent ◽  
Stability Theorems

Beam Vibrations With Time-Dependent Boundary Conditions

1950 ◽  
Vol 17 (4) ◽  
pp. 377-380
Author(s):  
R. D. Mindlin ◽  
L. E. Goodman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Time Dependent ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Vibration Problems

Abstract A procedure is described for extending the method of separation of variables to the solution of beam-vibration problems with time-dependent boundary conditions. The procedure is applicable to a wide variety of time-dependent boundary-value problems in systems governed by linear partial differential equations.

Nonlinear Phenomena in Axially Moving Beams with Speed-Dependent Tension and Tension-Dependent Speed

2021 ◽  
Vol 31 (03) ◽  
pp. 2150037
Author(s):  
Ling Chen ◽  
You-Qi Tang ◽  
Shuang Liu ◽  
Yuan Zhou ◽  
Xing-Guang Liu
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Fourier Transforms ◽  
Time Dependent ◽  
Nonlinear Phenomena ◽  
Maximum Lyapunov Exponent ◽  
Material Derivative ◽  
Domains Of Attraction ◽  
Moving Beam ◽  
Galerkin Truncation

This paper investigates some nonlinear dynamical behaviors about domains of attraction, bifurcations, and chaos in an axially accelerating viscoelastic beam under a time-dependent tension and a time-dependent speed. The axial speed and the axial tension are coupled to each other on the basis of a harmonic variation over constant initial values. The transverse motion of the moving beam is governed by nonlinear integro-partial-differential equations with the rheological model of the Kelvin–Voigt energy dissipation mechanism, in which the material derivative is applied to the viscoelastic constitutive relation. The fourth-order Galerkin truncation is employed to transform the governing equation to a set of nonlinear ordinary differential equations. The nonlinear phenomena of the system are numerically determined by applying the fourth-order Runge–Kutta algorithm. The tristable and bistable domains of attraction on the stable steady state solution with a three-to-one internal resonance are analyzed emphatically by means of the fourth-order Galerkin truncation and the differential quadrature method, respectively. The system parameters on the bifurcation diagrams and the maximum Lyapunov exponent diagram are demonstrated by some numerical results of the displacement and speed of the moving beam. Furthermore, chaotic motion is identified in the forms of time histories, phase-plane portraits, fast Fourier transforms, and Poincaré sections.

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